On monomial Burnside rings

Abstract

This thesis is concerned with some different aspects of the monomial Burnside rings, including an extensive, self contained introduction of the A−fibred G−sets, and the monomial Burnside rings. However, this work has two main subjects that are studied in chapters 6 and 7. There are certain important maps studied by Yoshida in [16] which are very helpful in understanding the structure of the Burnside rings and their unit groups. In chapter 6, we extend these maps to the monomial Burnside rings and find the images of the primitive idempotents of the monomial Burnside C−algebras. For two of these maps, the images of the primitive idempotents appear for the first time in this work. In chapter 7, developing a line of research persued by Dress [9], Boltje [6], Barker [1], we study the prime ideals of monomial Burnside rings, and the primitive idempotents of monomial Burnside algebras. The new results include; (a): If A is a π−group, then the primitive idempotents of Z(π)B(A, G) and Z(π)B(G) are the same (b): If G is a π 0−group, then the primitive idempotents of Z(π)B(A, G) and QB(A, G) are the same (c): If G is a nilpotent group, then there is a bijection between the primitive idempotents of Z(π)B(A, G) and the primitive idempotents of QB(A, K) where K is the unique Hall π 0−subgroup of G. (Z(π) = {a/b ∈ Q : b /∈ ∪p∈πpZ}, π =a set of prime numbers).